Airy Point Process via Supersymmetric Lifts

TL;DR

This paper develops a new method using supersymmetric lifts of Schur and Bessel functions to analyze the edge asymptotics of particle systems, revealing the emergence of the Airy point process.

Contribution

It introduces supersymmetric lifts as a novel analytical tool to study edge behavior in random matrix and representation theory models.

Findings

Explicit formulas for supersymmetric lifts of Schur and Bessel functions.

Asymptotic analysis demonstrating the appearance of the Airy point process.

Unified treatment of eigenvalue sums and tensor product signatures.

Abstract

We study the local asymptotics at the edge for particle systems arising from: (i) eigenvalues of sums of unitarily invariant random Hermitian matrices and (ii) signatures corresponding to decompositions of tensor products of representations of the unitary group. Our method treats these two models in parallel, and is based on new formulas for observables described in terms of a special family of lifts, which we call supersymmetric lifts, of Schur functions and multivariate Bessel functions. We obtain explicit expressions for a class of supersymmetric lifts inspired by determinantal formulas for supersymmetric Schur functions due to \cite{MJ03}. Asymptotic analysis of these lifts enable us to probe the edge. We focus on several settings where the Airy point process arises.